Banach Journal of Mathematical Analysis

On a reverse of Ando--Hiai inequality

Yuki Seo

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Abstract

In this paper, we show a complement of Ando--Hiai inequality: Let $A$ and $B$ be positive invertible operators on a Hilbert space $H$ and $\alpha\in [0,1]$. If $A\ \sharp_{\alpha}\ B \leq I$, then $$A^r \, \sharp_\alpha \, B^r \leq \|(A \,\sharp_\alpha \, B)^{-1}\|^{1-r} I \quad \text{for all }\, 0 <r \leq 1, $$ where $I$ is the identity operator and the symbol $\| \cdot \|$ stands for the operator norm.

Article information

Source
Banach J. Math. Anal., Volume 4, Number 1 (2010), 87-91.

Dates
First available in Project Euclid: 27 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1272374672

Digital Object Identifier
doi:10.15352/bjma/1272374672

Mathematical Reviews number (MathSciNet)
MR2593907

Zentralblatt MATH identifier
1186.47014

Subjects
Primary: 47A63: Operator inequalities
Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 47A64: Operator means, shorted operators, etc.

Keywords
Ando--Hiai inequality positive operator geometric mean

Citation

Seo, Yuki. On a reverse of Ando--Hiai inequality. Banach J. Math. Anal. 4 (2010), no. 1, 87--91. doi:10.15352/bjma/1272374672. https://projecteuclid.org/euclid.bjma/1272374672


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References

  • T. Ando and F. Hiai, Log-majorization and complementary Golden-Thompson type inequalities, Linear Algebra Appl. 197/198 (1994), 113–131.
  • R. Bhatia, Matrix Analysis, Springer, New York, 1997.
  • T. Furuta, Extension of the Furuta inequality and Ando–Hiai log-majorization, Linear Algebra Appl. 219 (1995), 139–155.
  • T. Furuta, Operator inequalities associated with Hölder-McCarthy and Kantorovich inequalities, J. Inequal. Appl. 2 (1998), 137–148.
  • T. Furuta, J. Mićić, J.E. Pečarić and Y. Seo, Mond-Pečarić Method in Operator Inequalities, Monographs in Inequalities 1, Element, Zagreb, 2005.
  • F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246(1980), 205–224.
  • R. Nakamoto and Y. Seo, A complement of the Ando–Hiai inequality and norm inequalities for the geometric mean, Nihonkai Math. J. 18 (2007), 43–50.